No derived residue map into structure cohomology of a normal surface modification

Given: A normal two-dimensional Noetherian local domain $A$ essentially of finite type over a field or complete equicharacteristic local base with residue field $\kappa$, and a projective normal modification $X\to \mathrm{Spec}A$.

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $\mathcal{F}$ all of whose members are nonempty, there exists a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for all $S\in \mathcal{F}$. (The Axiom of Choice)

def-dependent-choice. Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when $\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$ The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement. (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

def-normal-surface-modification-and-normalized-point-blowup. Normal schemes. A locally Noetherian scheme is normal if every local ring ${\mathcal{O}}_{X,x}$ is an integrally closed domain (def-normal-noetherian-ring). This is a local condition on the local rings and is checked on an affine open cover; it does not require the global section ring to be a domain. The empty scheme is normal vacuously. (Normal scheme modifications and normalized point blowups)

lem-local-normal-surface-modification-dimension-and-projective-cohomology. Assume AC and DC. Let $(A,\mathfrak{m})$ be a normal Noetherian local domain of dimension two and $f:X\to \mathrm{Spec}A$ an integral modification. Then $X$ has dimension two, all closed points have local dimension two, $f$ is an isomorphism off the closed point, $f_{\ast }{\mathcal{O}}_X={\mathcal{O}}_{\mathrm{Spec}A}$, and its special fibre has dimension at most one. (Dimension and cohomology of local normal surface modifications)

lem-projective-normal-surface-modification-h1-injects-off-special-fibre. Assume AC and DC. Let $A$ be a normal two-dimensional Noetherian local domain essentially of finite type over a field or complete equicharacteristic local base, and let $X\to \mathrm{Spec}A$ be a projective normal modification. If $U$ is the inverse image of the punctured spectrum, $H^1(X,{\mathcal{O}}_X)\to H^1(U,{\mathcal{O}}_U)$ is injective. (H1 of a normal surface modification injects off its special fibre)

lem-r-one-s-two-intersection-of-height-one-localisations. Assume the Axiom of Choice. If $R$ is a commutative Noetherian domain satisfying $(S_2)$, then inside its fraction field $K$ one has $R={\bigcap }_{\mathrm{ht}\mathfrak{p}=1}{R}_{\mathfrak{p}}$. For a field the empty intersection is interpreted as $K=R$. (r one s two intersection of height one localisations)

lem-normal-domain-implies-s-two. Assume the Axiom of Choice (The Axiom of Choice). Every commutative Noetherian integrally closed domain satisfies $(S_2)$. (normal domain implies s two)

thm-completion-of-a-noetherian-local-ring. Assume the Axiom of Choice. Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $\widehat{R}$ be its $\mathfrak{m}$-adic completion. 1. $\widehat{R}$ is a Noetherian local ring with maximal ideal $\mathfrak{m}\widehat{R}$. 2. The residue field is unchanged: $\widehat{R}\mathrm{/}\mathfrak{m}\widehat{R}\cong R\mathrm{/}\mathfrak{m}\mathrm{.}$ 3. The completion map $R\to \widehat{R}$ is faithfully flat. (Completion of a Noetherian local ring is local with the same residue field)